= Surface-electron and bulk-phonon heat-capacity crossover
{title2=$C_{\mathrm{surf}}/C_{\mathrm{bulk}}\propto A/(NT)$}
A neutral two-dimensional linear electronic spectrum has <heat capacity> proportional to $AT^2$, while three-dimensional acoustic <phonons> have <Debye heat capacity> proportional to $NT^3/\Theta_D^3$. Thus their ratio scales as $Ak_B^2\Theta_D^3/(\hbar^2v^2NT)$. At fixed <crystal> composition and shape, $A/N\propto1/L$ for linear size $L$, favoring the electronic surface contribution in smaller or colder specimens. The continuum laws require thermally populated wavelengths small compared with sample size; their formal low-temperature crossover need not lie inside that regime.
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